I'm not sure what your delta H is a reference to Astronut but it could be the energy absorbed when a bond is formed.

It's the net energy lost to break the bond, in this case, or the net energy lost to form it. If the graph were reversed it would be the net energy gained by breaking or forming it. When that breakage occurs you will still get a release of energy, there's no such thing as breaking or forming bonds without overcoming the activation energy needed to do so.

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Some reactions are endothermic. But this has nothing to do with the matter at hand.

It has everything to do with it, the process of breaking titanium atoms apart in a sample of titanium metal is an endothermic process.

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As for what bond energy actually is, take the following quote from Wikipedia:

You obviously don't know what "bond dissociation energy" is or you wouldn't have offered that quote just now.

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"Another example: an O–H bond of a water molecule (H–O–H) has 493.4 kJ mol-1 of bond dissociation energy,

Bond dissociation energy IS delta H, you just proved my point. From your own wiki page:
"For instance, the bond dissociation energy for one of the C-H bonds in ethane (C2H6) is defined by the process:
CH3CH2-H → Ethyl Radical + H.D0 = ΔH = 101.1 kcal/mol (423.0 kJ/mol)"
You're not talking about the energy actually required to start breaking the bond, you're talking about the energy actually lost in doing so, delta H on the above graph I posted.

"Every reaction in which bonds are broken will have a high energy transition state that must be reached before products can form. In order for the reactants to reach this transition state, energy must be supplied and reactant molecules must orient themselves in a suitable fashion. The energy needed to raise the reactants to the transition state energy level is called the activation energy, ΔE"Chemical Reactivity